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Summary

Since Lord Rayleigh introduced the idea of viscous damping in his classic work "The theory of sound" (1877) it has become standard practice to use this approach in dynamics covering a huge range of applications from aerospace to civil engineering. However, in majority of practical cases this approach is adopted more for mathematical convenience than modelling the physics of vibration damping. Over the past decade extensive research have been done on more general 'non-viscous' damping models and vibration of non-viscously damped systems. This book is the first comprehensive text to cover vibration problems with general non-viscous damping. The author will draw on his considerable research experience to produce a volume covering: non-viscously damped single and multi-degree of freedom systems; continuous systems with non-local and non-viscous damping; identification of non-viscous damping; computational aspects of non-proportional and non-viscous damping and sensitivity analysis. The text will be written from a vibration theory standpoint, with numerous worked examples relevant across a wide range of mechanical, aerospace and structural engineering applications.

Table of Contents

1 Introduction to Damping Models 1

1.1 Historical review of vibration damping 1

1.2 Damping in structural elements 1

1.2.1 Material damping effects 1

1.2.2 Non-material damping effects 1

1.3 Viscous damping models 1

1.3.1 Proportional viscous damping 1

1.3.2 Non-proportional viscous damping 1

1.4 Viscoelastic damping 1

1.5 Fractional damping 1

1.6 General nonviscous damping models 1

1.7 Damping in composite materials and structures

1.8 Nonlinear damping models 1

1.9 Damping enhancement in vibrating systems 1

1.9.1 Tuned mass dampers 1

1.9.2 Friction dampers 1

1.9.3 Impact dampers 1

1.10 Experimental identification of damping 1

1.11 Scope of the book 1

2 Review of Undamped and Viscously Damped Systems 3

2.1 Single-degree-of-freedom systems 3

2.2 Multiple-degree-of-freedom undamped systems 3

2.2.1 Modal analysis 3

2.2.2 Dynamic response 5

2.3 Proportionally damped systems 7

2.3.1 Condition for proportional damping 8

2.3.2 Generalized proportional damping 9

2.3.3 Dynamic response 10

2.4 Non-proportionally damped systems 20

2.4.1 Free vibration and complex modes 21

2.4.2 Dynamic response 24

2.5 Continuous systems 28

2.5.1 Axial vibration of rods 28

2.5.2 Flexural vibration of beams 28

2.5.3 Bending vibration of thin plates 28

3 Nonviscously Damped Single-Degree-of-Freedom Systems 29

3.1 Introduction 29

3.2 The equation of motion 30

3.3 Conditions for oscillatory motion 32

3.4 Critical damping factors 36

3.5 Characteristics of the eigenvalues 37

3.5.1 Characteristics of the natural frequency 37

3.5.2 Characteristics of the decay rate corresponding to the oscillating mode 40

3.5.3 Characteristics of the decay rate corresponding to the non-oscillating mode 43